Stability of three-dimensional dust acoustic waves in a dusty plasma with two opposite polarity dust species including dust size distribution

Propagation of dust acoustic waves (DAWs) with the effect of power law dust size distribution (DSD) in a magnetized dusty plasma with opposite polarity dust is studied. Using a reductive perturbation method, a Zakharov-Kuznetsov equation appropriate for describing three-dimensional DAWs is derived. The compressive and rarefactive solitons are possible in the present model. Due to the DSD effect, a soliton with a smaller amplitude and width and a larger velocity is observed. The stability criterion for obliquely propagating DAWs in such plasma using small-$k$ expansion method is investigated. The growth rate of instability is derived and analyzed under the effect of power law DSD. It is found that the growth rate of instability is strongly affected by the power law DSD. The relevance of these findings to space plasma phenomena is briefly discussed.

Figure 1

The dependence of the velocity ratio $\Lambda$ on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$ $u_o=1.1,$$\delta =11,$ ${\omega }_{cn}=0.05,$ $r_{n1}={10}^{-5},$ $r_{p1}=5\times {10}^{-4}$, and $\sigma =0.1$.

Figure 2

The dependence of the amplitude ${\varphi }_m$ of the compressive solitary wave on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$$l_{\eta }=0.7,$ $u_o=1.1,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 3

The dependence of the amplitude ${\varphi }_m$ of the rarefactive solitary wave on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$$l_{\eta }=0.7,$ $u_o=1.1,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 4

The dependence of the amplitude ratio $\Phi$ on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$ $u_o=1.1,$$\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 5

The dependence of the width $W$ of the compressive solitary wave on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$$u_o=1.1,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 6

The dependence of the width $W$ of the rarefactive solitary wave on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$$u_o=1.1,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 7

The dependence of the width ratio $\Delta$ on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 8

The variation of $S_i$ with the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 9

The $S_i=0$ surface plot via the variation of $R,\beta$, and $\delta$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7$, and ${\omega }_{cn}=0.05$.

Figure 10

The $S_i=0$ surface plot via the variation of $R,\beta$, and ${\omega }_{cn}$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7$, and $\delta =11$.

Figure 11

The dependence of the growth rate of the instability of DAWs $\Gamma$ on the dust radius ratio $R$ and the power law index $\beta$ for $l_{\zeta }=0.3,$$l_{\eta }=0.7,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 12

The dependence of the growth rate $\Gamma$ on $l_{\zeta }$ and $l_{\eta }$ for $R=3,$ $\beta =3.48,$ $\delta =11$, and ${\omega }_{cn}=0.05$.

Figure 13

The dependence of the growth rate $\Gamma$ on $R$ and ${\omega }_c$ for $l_{\zeta }=0.3,$ $l_{\eta }=0.7,$ $\delta =11$, and $\beta =3.48$.